Estimation of Dependence in Multivariate Extreme Value Statistics

In multivariate extreme value analysis, the extremal dependence structure between random variables can be characterized in several ways. A complete characterization can be obtained from the spectral measure or the stable tail dependence function. Alternatively, one can summarize the extremal dependency by a single number, known as the tail dependence coefficient. This coefficient measures the strength of the extremal dependence between the components of a bivariate random variable. Several well-performing estimators for the tail dependence coefficient have been introduced in the literature. However, these estimators do not take into account the presence of random covariates nor do they protect against potential contamination in the given data, which will adversely affect the quality of the estimation, especially when relevant data are scarce. To address these issues, we present a robust non-parametric method to estimate the conditional tail dependence coefficient using the minimum density power divergence criterion. The asymptotic properties of the estimator are studied under suitable regularity conditions.

In order to mitigate the impacts of extreme events, numerous risk measures have been introduced in the literature. In the multivariate context, the interest may be in the risk associated with one random variable when a related variable becomes extreme. The marginal expected shortfall was introduced to quantify such risk. Existing estimators for this measure do not take into account the presence of random covariates, which can be incorporated to improve the accuracy of the estimation. As such, we present an estimator for the conditional marginal expected shortfall. There are two main ingredients for this estimator: an in-sample estimator and an extrapolation method when the related variable is extreme. These are studied separately in two chapters, where the asymptotic normality of the proposed estimator in each chapter is established for a wide class of conditional bivariate distributions, with heavy-tailed conditional marginal distributions using empirical process arguments.
Among existing risk measures in the literature is the marginal mean excess, which is the expected excess of one risk above a high threshold conditional on a related variable exceeding another high threshold. In this thesis, we introduce a generalization of this measure in the regression setting, the so-called conditional marginal excess moment. We present an estimator for this new measure and establish its asymptotic normality under suitable conditions.

The performance of the estimator presented in each chapter will be evaluated by a simulation study. The efficiency and practical applicability will be illustrated on real datasets.